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A354325
Expansion of e.g.f. 1/(1 - x/4 * (exp(2 * x) - 1)).
3
1, 0, 1, 3, 14, 80, 558, 4522, 41864, 436032, 5046680, 64251176, 892361520, 13426491520, 217555171568, 3776935252560, 69942048682112, 1376150998836224, 28669321699355520, 630448829825395840, 14593473117397510400, 354696400190943197184, 9031466708133617225984
OFFSET
0,4
FORMULA
a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-3) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * k! * Stirling2(n-k,k)/(n-k)!.
MATHEMATICA
With[{nn=30}, CoefficientList[Series[1/(1-x/4 (Exp[2x]-1)), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Dec 02 2022 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/4*(exp(2*x)-1))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-3)*binomial(i, j)*v[i-j+1])); v;
(PARI) a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*stirling(n-k, k, 2)/(n-k)!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 24 2022
STATUS
approved